Questions on the use of algebraic techniques to prove geometric theorems.

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2answers
32 views

Find perpendicular vectors in subspace of $V_{3}$

Find all vectors of $V_{3}$ which are perpendicular to the vector $(7,0,-7)$ and belong to the subspace $L((0,-1,4), (6,-3,0)$. As a note, this is an extra question of a long exercise, the ...
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2answers
52 views

Prove $\sin^2 A = \sin^2 B \sin^2 C - 2\sin B \sin C \cos A$

I am asking for help with this proof: Given $\triangle ABC$. Prove that $\sin^2 A = \sin^2 B+ \sin^2 C - 2\sin B \sin C \cos A$
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2answers
63 views

$\overline{x} \times \overline{a} = \overline{b}$ has a solution when $ \langle\overline{a},\overline{b} \rangle =0$

I'm trying to solve this exercise: Let $\overline{a} \neq \overline{0}$, $\overline{b}$ be two vectors of the Euclidean vector space $V_{3}$. Prove the equation $\overline{x} \times \overline{a} = ...
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1answer
18 views

How can I find the coordinates of a point which is the reflection of a point about a line in 3D

I am currently working on a project on Matlab and I need to find the coordinates of a point which is reflected about a line. I know how to do it in 2D but in 3D things are getting ugly. So, we have a ...
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1answer
35 views

Scalar times Point + Scalar times Point?

Let $P$, $Q$ be a pair of points in the Euclidean plane and let $t_1$, $t_2$ be a pair of scalars. My textbook says that the following operations are nonsense: $$P + Q\\ t_1 \cdot P$$ However $t_1 ...
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1answer
73 views

How to find the equation of a parabola with vertex on the line y = -3x?

Its axis are parallel to the y-axis and passing through (-7,13) and (5,1).
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2answers
56 views

Find the tangents to the following curve from the given point.

2x^2 + y^2 = 54 from (10,1) P.S. I still don't study calculus. This lesson is from analytic geometry and I have no idea how to solve it because my professor didn't teach it. So if someone could tell ...
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1answer
54 views

A detailed basic-level explanation of equations of lines and planes in 3-d geometry

I've searched multiple blogs but couldn't find anything helpful for my level. (I'm in the 12th grade learning about vectors in maths). I basically need some thorough explanations regarding plane and ...
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1answer
21 views

points in the 3-dimensional space

Let $A=(a,b,c)$, $B=(d,e,f)$ and $C=(g,h,i)$ be points in the $3$-dimensional real vector space. It is well known that we can consider a new referential where we can see these points as ...
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0answers
84 views

Knight's metric: ellipse and parabola.

Knight's metric is a metric on $\mathbb{Z}^2$ as the minimum number of moves a chess knight would take to travel from $x$ to $y\in\mathbb{Z}^2$. What does a parabola (or an ellipse) became with this ...
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0answers
9 views

What's the point in the general form of a plane equation?

So I've been reading up on my maths (Mathematics for Computer Graphics by John Vince FYI) and come to analytic geometry and I have a question. Why define the Cartesian form of a plane equation as: $$ ...
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1answer
33 views

exercise with lines algebra

This is an exercise I really can't solve by myself. 1) Let A(1,-1,0) a point on a line (e) line 2) Let (d) be a line perdicular to (e), given by a parametric equation. How I can find the equation ...
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0answers
29 views

points of intersection of two circles drawn on a sphere

How does one write and thereafter solve the equations to find the points of intersection of two overlapping circles drawn on the surface of a sphere? I am looking for a simple understandable solution ...
2
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2answers
99 views

check if point is on a plane (using Heron formula ?)

Is this true that if any of parameters a, b, c, d is equal to sum of three others then 4 points are on same plane? I am given 4 points in 3 dimensional space. Is this correct to state that all 4 ...
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2answers
45 views

Three coplanar vectors - proving a statement

Is it true that three vectors $\vec{u}, \vec{v} , \vec{w} $ lie on the same plane if and only if there exists constants $A,B,C,D$ for which $A\vec{u} + B\vec{v} + C\vec{w} +D =\vec{0} $ ? If so, how ...
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1answer
33 views

Analytic geometry and calculus question

The tangent to the curve $ x^{0.5} + y^{0.5} = a^{0.5}$ ,$(a>0)$ intersects the $x$ axis in point $M_1$ , and the $y$ axis in point $M_2$. Prove that for any point on the curve $|OM_1|+|OM_2| = ...
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2answers
30 views

Analytic geometry and calculus combined question

Show that the equation for the tangent with the slope $m$, $(m≠0)$ to the parabola $y^2 = 4px$ is $y = mx + \frac{p}{m}$. How this is done? What is the method for proves of this kind?
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2answers
41 views

Finding root for the segment - found the formula but it doesn't work for some values - wrong formula?

I have the segment, defined as $(x_1, y_1)$, $(x_2, y_2)$. I know that $y_1\ge 0$ and $y_2 < 0$. I want to compute the root point for that segment. I decided to do it that way: ...
1
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1answer
210 views

How to find out if four points are on the same plane, only by using distances?

There is a method called Cayley-Menger determinant in order to find if 3 points are collinear, 4 points are coplanar etc. provided that all the pairwise distances are given. However, in 2-D, there is ...
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3answers
53 views

Analytic geometry and calculus mixed question

Find the normal equation to the graph $ y = 2x^2$, that goes through the point $ (1,0.25)$. How this is done? Im not even really sure what I'm being asked here...
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0answers
42 views

Rotation of axes transformation as definition of vectors

Given a three-axes coordinate system ${1,2,3} $ by the right-hand rule, and a new coordinate system ${1',2',3'}$ , I know that one can define a vector $\vec{x}$ to be something that obeys the ...
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3answers
66 views

Calculus and analytic geometry question

Find the tangent of the angle in which the functions $x^3 $, and $x^2 $ intersect $(x≠0)$ . I find this question to be quite funny since the intersection point has two tangents going to it, with ...
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2answers
67 views

(I only need some hints)Find the vector equation of a space curve that represents an ellipse with the given center that lies in the given plane

Full disclosure, this is for a Calculus III graded homework set--though we are allowed to use any resources available to complete it. I feel I have a good understanding of space curves, though my ...
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3answers
28 views

solving for the left side for a double angle formula

$\cos t-\sin 2t=0$ Solve the left hand side so that it equals zero. Do I use $(2\sin t \cos {t})$ for $\sin 2t$?
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1answer
158 views

Does a square have an equation? [duplicate]

can you model a square in an equation ? like a circle for example $r^2 = x^2 + y^2$ and lets say we have a square with: centered at $(3,3)$ $2 \leq x \leq 4$ and $2 \leq y\leq 4$ can we somehow ...
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2answers
63 views

Finding the point that a normal line goes through

I have been stumped on a homework problem for quite some time and I'm hoping to get some help with it. The line from the origin to the point $(a, f(a))$ on the graph of $f(x) = \frac1{x^2}$ is ...
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1answer
37 views

How to Create a Plane Inside A Cube

I have a $e \times e \times e$ cube and I want to create random planes with equation $ax + by + cz + d = 0$ inside this cube. I will put random points on those randomly created planes as well. Here ...
2
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1answer
71 views

Proof for $\cos(\alpha)^2 + \cos(\beta)^2 + \cos(\gamma)^2 = 1$ in Euclidean space

What is the proof for this formula: $$ \cos(\alpha)^2 + \cos(\beta)^2 + \cos(\gamma)^2 = 1, $$ where $\alpha$, $\beta$ and $\gamma$ are the angles between a vector and the base of a right-handed ...
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1answer
25 views

Graph a line oriented by an specific angle?

I'm writing a software that plots gps data on a map, and so far it has been riddled with complex math problems, many of which I was able to fix by myself but this one I can't figure out. The software ...
2
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0answers
44 views

Cauchy-Schwarz on a Euclidian Space

I was thinking about this proof of the cauchy-schwarz inequality, I wanna show that $$|\langle u,v\rangle|\leq|u||v|$$. We know that, $$|\langle u,v\rangle| = ||u||v|\cos{\theta}|$$ where $\theta$ ...
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4answers
112 views

surface that is created by the intersection of paraboloid and plane

Find the surface that is created by the intersection of the paraboloid $x^2+y^2-z=0$ and the plane $z=2$. $$x^2+y^2-z=0 \Rightarrow x^2+y^2=z$$ $$z=2$$ EDIT: I had to find the area of the surface ...
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1answer
36 views

locus sections and circles--symmetry

A. Let L = {(x,y,z)|the distance from (x,y,z) to the y-axis is 6}. Describe what shape L is. So far, I have that it's a circle, but I'm not sure how to describe it fully. Would it be a circle that ...
1
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1answer
10 views

Finding the degree to cover a flare shape

I have a lamp shade frame that I want to cover with bamboo slats. The top is 12" around and the bottom is 33" around. The distance from the top to the bottom most part is 6". What degree, with minor ...
2
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1answer
89 views

Interesting Analytic Geometry Problem: Find internal angles given coordinates of final point and length of line segments

I have been mulling over a really interesting question in analytic geometry that is much harder than it first appears to be. Hope you can provide some insight into solving it. If you only know: ...
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1answer
77 views

Differential geometry question.

Please explain how to solve this question. Thank you:) And sorry for hand-writing.
2
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2answers
67 views

“Looping” equation

I'm looking for a equation that describes the shape of a "Looping" in the best way. I really don't know how to start here, as it isn't even a function (if it were, I could just use spline ...
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1answer
63 views

Rotate the Points on a Plane $P = ax+by+cz + d = 0$ parallel to $z = 0$ plane

I have a plane $P = ax+by+cz + d = 0$ and many points on that plane. I want to rotate $P$ so that it becomes parallel to $z = 0$ plane. Which method should I use? I know that the normal vector of my ...
1
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1answer
25 views

Find Points in The Cube

In the cube $ABCDA'B'C'D'$, we have $3\overrightarrow{AM}=\overrightarrow{MD}$ and $2\overrightarrow{D'N}=\overrightarrow{NB'}$. Find the points $M$ and $N$ in the cube; So, i can't find a way to ...
1
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0answers
21 views

finding a locus in a $3$ dimensions

Given a Tetrahedron $OABC$ such that $O(0,0,0),A(a,0,0),B(0,b,0),C(0,0,c)$ ; $a,b,c$ are not zero. We build a plane $\pi$ that is parallel to $z$-axis and also to $AB$. Plane $\pi$ cuts plane $ABC$ ...
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2answers
215 views

The number of grid points near a circle.

There is a circle with center $(0, 0)$ and radius $r$. Let $n$ be the number of grid points inside or on the circle that at least one of its neighboring (up, down, left, right) grid points is outside ...
2
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1answer
160 views

How To Generate Random Points on the Positive Side of a Plane in 3-D

Edit: The question can also be interpreted as: How to generate random coplanar points in a cube? Here is what I am struggling with: I have a cube, whose origin is $(0,0,0)$ and one edge length ...
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3answers
81 views

Trigonometry, knowing 3 sides how to find the height?

I have a mathematician problem where, I knew the 3 sides of a triangle, with these sides I can figer out what type of type of triangle is. What I realy want to find is the height of the triangle and ...
0
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1answer
18 views

Concept of parallelism in analytic terms

Below I cited a passage from Apostol's Calculus. I don't understand how to use the identity to show that two lines with equal slopes are parallel. Concepts such as perpendicularity and parallelism ...
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2answers
41 views

This question on right triangles. [closed]

a ladder 10 ft. in length is leaning against a wall 8 ft. above the ground. a 6ft. guy tries pass underneath the ladder 1 ft. away from the wall. will he be able to pass underneath the ladder.
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2answers
23 views

How will I get the point of intersection?

I'm confuse on how will I get the point of intersection of these two equations: $x^2+y^2+5x+y-26=0$ and $x^2+y^2+2x-y-15=0$ I tried using the elimination method but I can't get it.
0
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2answers
293 views

find the equation of the locus of a moving point which is always equidistant from the y-axis and the point (-6,4)

Do you know how to solve its equation? Already solved some locus problems that gives points but not in the y or x axis problems.
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1answer
28 views

Formula to plot a non-linear graph

Firstly, thank you very much in advance. I need to express a non-linear graph comprised (piecewise) of the following linear elements: A line from $(x=0,y=100)$ to $(x=10,y=100.5)$ A line from ...
1
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1answer
29 views

A rectangle $OACB$ with two axes as two sides,the origin $O$ as a vertex is drawn in which the length $OA$ is four times the width $OB$…

A rectangle $OACB$ with two axes as two sides,the origin $O$ as a vertex is drawn in which the length $OA$ is four times the width $OB$.A circle is drawn passing through the points BC and touching ...
0
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2answers
103 views

Equation of the Circle

How to find the equation of a circle if the givens are the: Case 1: Tangent to $2x + 3y + 13 = 0$ and $2x - 3y - 1 = 0$; contains $(0,4)$ Case 2: Tangent to $x - 3y - 7 = 0$ and $3x + y - 21 = 0$; ...
1
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1answer
43 views

analytic geometry , Orthogonal projection

II) take the point P = (2,1,0) , the line r : X = (0,0,0) + t(2,1,0) and the plane B: x + y + z -3 = 0 . For each point Q let it be Q' his orthogonal projection to B .Find the Q points that outputs ...